Self-interacting dark matter implied by nano-Hertz gravitational waves

The self-interacting dark matter (SIDM) paradigm offers a potential solution to small-scale structure problems faced by the collision-less cold dark matter. This framework incorporates self-interactions among dark matter particles, typically mediated by a particle with a MeV-scale mass. Recent evidences of nano-Hertz gravitational waves from pulsar timing arrays (PTAs) such as NANOGrav, CPTA, EPTA, and PPTA suggest the occurrence of a first-order phase transition (FOPT) at a MeV-scale temperature. Considering the close proximity between these two scales, we propose that the mediator mass in the SIDM model originates from the spontaneous breaking of a $U(1)'$ symmetry, which is driven by the FOPT indicated by PTA data. Consequently, the alignment of these two scales is believed to be deeply connected by the same underlying physics. By extensively exploring the parameter space, remarkably, we find that the parameter space favored by SIDM just provides an explanation for the PTA data.


I. INTRODUCTION
The widely accepted cold dark matter (CDM) model successfully explains the Universe's structure and evolution.However, it faces challenges when addressing smallscale structure problems [1][2][3][4][5][6][7][8].These difficulties arise in understanding the behavior and distribution of dark matter (DM) within galaxies and galaxy clusters.To address these challenges, the self-interacting dark matter (SIDM) paradigm has emerged.It suggests that dark matter particles can interact through short-range forces mediated by a new particle called "dark photon" or "dark mediator," typically with a mass at the MeV range.Validating the SIDM paradigm requires crucial searches for evidence of the existence of the dark sector.
The remarkable proximity between the scale of mediating DM self-interaction and that of the FOPT indicated by the PTA data suggests a profound connection between them.We propose that the SIDM mediator mass originates from the spontaneous symmetry breaking of a new gauge sector, which occurs through a strong FOPT in the early Universe.Consequently, the alignment of these two scales signifies the presence of a unified underlying physics that governs the dynamics within the dark sector.
In this work, we present a concise and comprehensive model within the framework of SIDM to explain the recently reported PTA data, specifically focusing on the NANOGrav and CPTA observations.By thoroughly investigating our proposed model, we explore and identify a parameter space that can successfully account for the observed GW signals while addressing the smallscale structure problems.This model holds promise for experimental testing in the near future.
The article is organized as follows: Section II provides an introduction to our model and explores the physics associated with SIDM.Section III focuses on the calculation of the FOPT and the corresponding GW signals.In Section IV, we present our numerical results, showcasing the viable parameter space, and engage in further discussions.Finally, we conclude in Section V.

II. SELF-INTERACTING DARK MATTER
Small-scale problems are series of discrepancies between astrophysical observations and the simulation of arXiv:2306.16966v2[hep-ph] 2 Jul 2023 collision-less CDM at the scale smaller than O(1) Mpc.For example, the diversity of inner rotation curves of spiral galaxies is difficult to be explained by CDM [4].However, if there is an elastic scattering process between DM particles, the inner region of DM halo will be heated up and avoid being too dense due to thermalization, then the diversity problem can be alleviated [31][32][33].
Overall, for dwarf galaxies or galaxies with a DM average velocity v around 10 − 200 km/s, the cross section σ/m DM needs to be within the range O(1)−O(10) cm 2 /g [34,35].However, for galaxy clusters where v ∼ 2000 km/s, current fit result favors σ/m DM ∼ 0.2 cm 2 /g [36].It has been shown that a velocitydependent σ/m DM can solve small-scale problems at different systems .Such a velocity-dependent cross section can be easily induced via a light mediator between DM particles [46].
In this work, we propose a model that the mediator A is the gauge boson of a spontaneous broken dark U (1) symmetry.The breaking of this U (1) is through a FOPT in the elarly Universe, which generates the 10 −8 Hz GWs, as we will show later.The corresponding Lagrangian for the dark sector is given by where χ is the Dirac DM candidate, and S = (φ+iη)/ √ 2 is the dark Higgs field whose potential is Both χ and S carry unit charge under U (1) , and D µ ≡ ∂ µ + ig A µ with g being the dark gauge coupling.The Mexican hat shape of potential (2) generates a nonzero vacuum expectation value v s = µ/ √ λ, breaking the U (1) spontaneously, resulting in a massive A with m A = g v s and dark Higgs boson φ with m φ = √ 2µ.Previous studies show that a DM candidate with a mass m χ ∼ O (10) GeV and a mediator with a mass m A ∼ O (10) MeV can well fit the DM data from dwarf galaxies to galaxy clusters [58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76].In our model, for v s ∼ O(10) MeV and g ∼ O(1), a mediator mass m A around 10 MeV can be easily generated.On the other hand, the scale of v s also indicates the phase transition occurs at a temperature around MeV.
To fit the small-scale data, one needs to calculate the thermal averaged DM scattering cross section σ [77].Here σ is given by where σ V is the viscosity cross section [71,78,79] and v 0 is the velocity dispersion.Analytic formulae of DM scattering cross section at different parameter regions corresponding to different coupling strength and kinetic energy have been given in the literature [59,71,77,[80][81][82][83], and implemented in public package CLASSICS [77], which is used in our calculation.The fit result will be given in Section IV.
Here we also assume the chemical potential in the dark sector is small enough to not affect the FOPT of the dark sector.

III. FIRST-ORDER PHASE TRANSITION FROM DARK SECTOR
Shortly after the inflationary reheating, the Universe enters the radiation era with high temperature and density.We assume the dark and visible sectors are in thermal equilibrium (see Section IV).The scalar potential receives thermal corrections and becomes temperature dependent.The one-loop finite temperature potential reads where V 0 is Eq. ( 2) with S → φ/ √ 2, V 1 is the Coleman-Weinberg potential, δV is the counter term, V T 1 is the thermal correction, and V daisy is the daisy resummation.The complete form of V T (φ, T ) is given in Appendix A.
At zero temperature, the U (1) symmetry is spontaneously broken.However, at high temperatures in the early Universe, thermal corrections can restore the U (1) symmetry.This is evident from V T (φ, T ) ≈ (−µ 2 + g 2 T 2 /4)φ 2 /2 at φ ∼ 0, where a positive mass square term arises due to sufficiently high temperatures.Initially the Universe is at φ = 0.As it cools down and the potential shape changes, it transitions to the vacuum state φ = 0.This transition can occur smoothly through the rolling of the φ field.However, in certain parameter ranges, a potential barrier induced by the A field separates the two vacua, leading to a discontinuous quantum tunneling process known as a FOPT that proceeds via vacuum bubble nucleation and expansion.
To quantitatively calculate the FOPT dynamics, one needs to solve the O(3) symmetric bounce solution for V T and derive the classical action S 3 , and decay probability per unit volume is then [97] The nucleation of bubbles containing the true vacuum occurs at a temperature T n when the decay probability within a Hubble volume and a Hubble time, given by Γ(T n )H −4 (T n ), reaches O(1).Note that bubble nucleation itself does not guarantee the successful completion of a FOPT.A more rigorous criterion of FOPT is the existence of percolation temperature T p at which the true vacuum bubbles form an infinite connected cluster, and the volume of space occupied by the true vacuum keeps increasing [98].However, for a mild FOPT (as considered here), which ensures completion once nucleation occurs, we adopt the nucleation condition as a practical criterion and rewrite it as where M Pl = 1.22 × 10 19 GeV is the Planck scale, and g * is the number of relativistic degrees of freedom.We focus on T n ∼ MeV, where g * = 10.75 and the right-hand-side of Eq. ( 6) is around 190.During a FOPT, the collision of bubbles, the motion of sound waves and turbulence in the plasma generate stochastic GWs [99].Typically, the bubble walls reach a terminal velocity due to the friction force exerted by the plasma particles, and most of the released vacuum energy goes to surrounding plasma.As a consequence, the main GW sources are sound wave and turbulence, and the former usually dominates [100].The GW spectrum today, defined as with ρ GW and ρ c being the GW and current Universe energy density, respectively, can be written as numerical functions of the FOPT parameters {α, β/H * , T * , v w } [101], where α is the ratio of latent heat to the radiation energy density, β/H * is the inverse ratio of the FOPT duration to the Hubble time scale, T * is the temperature and v w is the bubble velocity.The sound wave contribution is [102] Ωsw(f )h 2 = 5.71 × 10 −8 vw κV α where κ V is the fraction of vacuum energy that is released to bulk motion, and is the peak frequency, which implies f sw ∼ 10 −8 Hz for T * ∼ MeV.
To obtain the GW spectrum, we use the package CosmoTransitions [103] to resolve T n , and analyze the hydrodynamic motion of the plasma using the bag model to determine the FOPT energy budget [104].Numerical simulations from previous studies are then utilized to calculate Ω GW (f )h 2 contributions from sound waves and turbulence, where T * = T n , H * = H(T n ) and v w = 0.9 are adopted.We account for the finite lifetime of sound waves through an additional suppression factor Hτ sw 1 [105].While turbulence usually has a subleading contribution, it modifies the tail of the sound wave GW signal, thus we include its effect in our analysis using results from Ref. [106].

IV. RESULTS AND DISCUSSION
Using the established framework, we conducted numerical calculations to determine the parameter space necessary to address both the small-scale problems and explain the PTA data.By systematically exploring this parameter space, we identified specific benchmark points (BPs) that simultaneously reconcile the PTA data and resolve the small-scale problems, as summarized in Table I.In the left and right panels of Fig. 1, we present the velocity-dependent elastic cross-sections of DM particles and the corresponding GW spectra derived from the BPs, respectively.As we explained in Section II, in the velocity region 10 − 200 km/s, we require the cross section σ/m χ to be within range O(1) − O(10) cm 2 /g [34,35].For galaxy groups where v ≈ 1150 km/s and galaxy clusters where v ≈ 1900 km/s, we use the result given in [36] which requires σ/m χ = 0.5 ± 0.2 cm 2 /g and σ/m χ = 0.19±0.09cm 2 /g, respectively.The NANOGrav data are from the collaboration result [9], while the CPTA data point is converted from the best fit point of f = 14 nHz and lgA = −14.4+1.0 −2.8 [10].Remarkably, we find that the identified BPs not only satisfy the required cross section values to explain the small-scale problems but also accommodate the observational data from NANOGrav and CPTA.

a t e x i t s h a 1 _ b a s e 6 4 = " X k X W c K f l S 3 4 I X n x I m d q K D B t n 1 N 0 = " > A A A C A H i c b Z C 7 T s M w F I a d c i v l F m B g Y L G o k J h K U p B g r G B h L B K 9 S E 2 o H N d p r d p J Z J 8 g q i g L r 8 L C A E K s P
r e r O e r B f r 3 f q Y t h a s 2 c w + + i P r 8 w e W v J Z q < / l a t e x i t > 0 .1 cm   Since the FOPT temperature is relatively low, a few discussions on other cosmological constraints are in order.As T n 3 MeV, the CMB and BBN constraints can be satisfied; for example, the effective number of neutrinos (N eff ) is essentially not affected by the FOPT reheating [107,108].The constraint from ultra-compact minihalo (UCMH) abundance is rather severe [109,110], but this issue can be relaxed by choosing a conservative value of the red shift of the last formation of UCHM at z c = 1000.
We further require the lifetime of φ to be short enough to avoid energy injection to the visible sector during the BBN period.For particles with mass larger than about 2 MeV and an e + e − decay final state, the lifetime should be shorter than 0.1 s, assuming the dark sector number of degrees of freedom at MeV is 1 [111].In our model, the decay width of φ is suppressed by its tiny mixing angle with the SM Higgs boson, which is constrained to be |θ| 10 −5 by the Higgs exotic decay [112], making the lifetime exceed 10 s.However, this bound can be avoided by introducing high dimensional operators such as S † Sē R H † L/Λ 2 with Λ 10 2 TeV.Also note that this operator can keep the dark sector in thermal equilibrium with the visible sector for Λ 10 TeV.

V. CONCLUSION
In this study, we established that a FOPT at the MeV scale, as inferred from the PTA data, provides a compelling mechanism for generating a mediator mass within the MeV range, which is crucial in addressing the small-scale problem within the framework of the SIDM.By considering various constraints, we identified the parameter space where the small-scale structure problem can be effectively resolved while simultaneously fitting the PTA data.
Our model is testable in the near future.To ensure compliance with the constraints from BBN and the CMB, as well as to maintain thermal contact between the dark and visible sectors, our model requires a dimension-6 operator S † Sē R H † L/Λ 2 with Λ 10 TeV, which induces the φe + e − and φφe + e − couplings and hence can be probed at the colliders.For example, LEP has constrained Λ 500 GeV by the e + e − → γφφ, while the future 1 TeV ILC can reach Λ ∼ TeV [113].The Higgs exotic decay h → φφ also provides a good probe to the scalar-Higgs portal coupling, which can be searched at the LHC [112].
In summary, the PTA data can help us better reveal the properties of SIDM by combing with those astronomy observations and shed light on the nature of DM.The upcoming experiments have the potential to detect various signals from our model, providing a complementary investigation of the new physics associated with nano-Hertz GWs.In this appendix we list the concrete expression of each term in Eq. ( 4).The Coleman-Weinberg potential can be derived using the field-dependent mass as where M 2 φ (φ) = −µ 2 + 3λφ 2 , M 2 η (φ) = −µ 2 + λφ 2 , M A (φ) = g φ, and µ R is the renormalization scale which is adopted as 10 MeV.We have dropped the contribution from the Goldstone η to avoid IR divergence [135].The counter term δV (φ) = − δµ 2 2 φ 2 + δλ 4 φ 4 , (A1) is derived by the conditions such that the zero temperature tree-level relations between (µ 2 , λ) and (v s , m A , m φ ) still hold.The one-loop thermal correction is where n φ,η = 1 and n A = 3, and the Bose thermal integral is defined as Finally, the daisy resummation term is where we only consider the dominant A contribution, as the parameter space of interest always has g 2 λ.The above discussions define our finite temperature potential V T (φ, T ) in Eq. ( 4).
The potentials of the four BPs described in the main text are ploted in Fig. 2, where the solid lines represent V T at critical temperature T c at which two dengnerate vacua exist, while dashed lines represent V T at nucleation temperature at which the bubbles start to emerge.We can clearly see that there is a barrier separating the two local minima of the potential, and the dominant contribution to this barrier is the A from thermal loop.At T n < T c , the φ = 0 minimum has a lower energy that it becomes the true vacuum, and the FOPT happens through the decay from φ = 0 to φ = 0. We obtain ∆φ/T n ∼ 5 for the BPs, as the observed GW signal requires a strong FOPT.This also implies that the hightemperature expansion is not suited in this scenario, thus we include the complete expression to calculate the FOPT.
hvi [km/s] < l a t e x i t s h a 1 _ b a s e 6 4 = " d w p W + j 3 8 E x O A T o X Z 2 5 E y / q n 9 k J A = " > A A A C C X i c b V A 9 S w N B E N 3 z 2 / h 1 a m m z G A S r e K e C F i J B G 0 s F k w h 3 R 9 j b z C V L 9 j 7 Y n Q u G I 6 2 N f 8 X G Q h F b / 4 G d / 8 Z N v M K v B w O P 9 2 a Y m R d m U m h 0 n A 9 r a n p m d m 5 + Y b G y t L y y u m a v b z R 1 m i s O D Z 7 K V N 2 E T I M U C T R Q o I S b T A G L Q w m t s H 8 + 9 l s D U F q k y T U O M w h i 1 k 1 E J D h D I 7 V t 6 k u I 8 I Q O q K 9 E t 4 e n 1 K e e j 3 C L R T / e 0 6 O g b V e d m j M B / U v c k l R J i c u 2 / e 5 3 U p 7 H k C C X T G v P d T I M C q Z Q c A m j i p 9 r y B j v s y 5 4 h i Y s B h 0 U k 0 9 G d M c o H R q l y l S C d K J + n y h Y r P U w D k 1 n z L C n f 3 t j 8 T / P y z E 6 D g q R Z D l C w r 8 W R b m k m N J x L L Q j F H C U Q 0 M Y V 8 L c S n m P K c b R h F c x I b i / X / 5 L m v s 1 9 6 C 2 f 3 V Y r Z + V c S y Q L b J N d o l L j k i d X J B L 0 i C c 3 J E H 8 k S e r X v r 0 X q x X r 9 a p 6 x y Z p P 8 g P X 2 C W i I m Y c = < / l a t e x i t > 1 0 cm 2 / g < l a t e x i t s h a 1 _ b a s e 6 4 = " I B / L X D d g u s o p L b Y W 0 c X E m + k t e j 0 = " > A A A C A X i c b Z D L S s N A F I Y n X m u 9 R d 0 I b g a L 4 K o m V d B l 0 Y 3 L C v Y C T S y T 6 a Q d O p O E m R O x h L j x V d y 4 U M S t b + H O t 3 F 6 W W j r D w M f / z m H M + c P E s E 1 O M 6 3 t b C 4 t L y y W l g r r m 9 s b m 3 b O 7 s N H a e K s j q N R a x a A d F M 8 I j V g Y N g r U Q x I g P B m s H g a l R v 3 j O l e R z d w j B h v i S 9 i I e c E j B W x 9 5 3 H Q 9 7 w B 4 g o z K / q 5 x M u J d 3 7 J J T d s b C 8 + B O o Y S m q n X s L 6 8 b 0 1 S y C K g g W r d d J w E / I w o 4 F S w v e q l m C a E D 0 m N t g x G R T P v Z + I I c H x m n i 8 N Y m R c B H r u / J z A Y b b 4 N 7 G a D l l y x 9 + s 8 5 O j 5 / k A i u w X G + r c L S 8 s r q W n G 9 t L G 5 t b 1 j 7 + 4 1 d Z w q y h o 0 F r F q B 0 Q z w S P W A A 6 C t R P F i A w E a w X D 6 3 G 9 9 c C U 5 n F 0 B 6 O E + Z L 0 I x 5 y S s B Y X f v A 9 b A H 7 B E y K v P 7 6 u m U + 3 n X L j s V Z y K 8 C O 4 M y m i m e t f + 8 n o x T S W L g A q

2
/ g < l a t e x i t s h a 1 _ b a s e 6 4 = " l w b r M F V a F O M K O F E A l 1 d Z q W M H q d 8 = " > A A A C A n i c b Z D L S s N A F I Y n X m u 9 R V 2 J m 8 E i u I p J F X R Z d O O y g r 1 A E 8 t k O m m H T i 7 M n I g l B D e + i h s X i r j 1 K d z 5 N k 7 b L L T 1 h 4 G P / 5 z D m f P 7 i e A K b P v b W F h c W l 5 Z L a 2 V 1 z c 2 t 7 b N n d 2 m i l N J W Y P G I p Z t n y g m e M Q a w E G w d i I Z C X 3 B W v 7 w a l x v 3 T O p e B z d w i h h X k j 6 E Q 8 4 J a C t r r l v W 4 6 L X W A P k N E w v 6 u e T L m f d 8 2 K bd k T 4 X l w C q i g Q v W u + e X 2 Y p q G L A I q i F I d x 0 7 A y 4 g E T g X L y 2 6 q W E L o k P R Z R 2 N E Q q a 8 b H J C j o + 0 0 8 N B L P W L A E / c 3 x M Z C Z U a h b 7 u D A k M 1 G x t b P 5 X 6 6 Q Q X H g Z j 5 I U W E S n i 4 J U Y I j x O A / c 4 5 J R E C M N h E q u / 4 r p g E h C Q a d W 1 i E 4 s y f P Q 7 N q O a d W9 e a s U r s s 4 i i h A 3 S I j p G D z l E N X a M 6 a i C K H t E z e k V v x p P x Y r w b H 9 P W B a O Y 2 U N / Z H z + A H h F l t w = < / l a t e x i t > 0 .0 1 cm 2 / g < l a t e x i t s h a 1 _ b a s e 6 4 = " L 8 e B b R y h c z 9 K / u e w g g H 9 8 C O v m H k = " > A A A C A 3 i c b Z D L S s N A F I Y n X m u 9 R d 3 p Z r A I r m p S B V 0 W 3 b i s Y C / Q x D K Z T t q h M 0 m Y O R F L C L j x V d y 4 U M S t L + H O t 3 F 6 W W j r g Y G P / z + H M + c P E s E 1 O M 6 3 t b C 4 t L y y W l g r r m 9 s b m 3 b O 7 s N H a e K s j q N R a x a A d F M 8 I j V g Y N g r U Q x I g P B m s H g a u Q 3 7 5 n S P I 5 u Y Z g w X 5 J e x E N O C R i p Y + 8 7 Z c f 1 s A f s A T I q 8 7 v K y Y R 7 e c c u G X N c e B 7 c K Z T Q t G o d + 8 v r x j S V L A I q i N Z t 1 0 n A z 4 g C T g X L i 1 6 q W U L o g P R Y 2 2 B E J N N + N r 4 h x 0 d G 6 e I w V u Z F g M f q 7 4 m M S K 2 H M j C d k k B f z 3 o j 8 T + v n U J 4 4 W c 8 S l J g E Z 0 s C l O B I c a j Q H C X K 0 Z B D A 0 Q q r j 5 K 6 Z 9 o g g F E 1 v R h O D O n j w P j U r Z P S 1 X b s 5 K 1 c t p H A V 0 g A 7 R M X L R O a q i a 1 R D d U T R I 3 p G r + j N e r J e r H f r Y 9 K 6 Y E 1 n 9 t C f s j 5 / A O r m l x Y = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " c 9 a T p A K B T K b F 8 H y 0 s v l l 6 + 2 7 5 f W N l 9 c P H t e b 6 x p m 3 p R P Q F V Z Z d 5 F x D 0 o a 6 K J E B R e F A 6 4 z B e f Z + H j q n 1 + B 8 9 K a 7 z g p I N U 8 N 3 I o B c c g 9 Z u n z A Z 7 m q 6 Y l 7 n m N V P c 5 A r o F W V u x n Z 0 v 2 J i J G v K K E O 4 x q o n d P t y t 5 2 3 L 6 s v S Z D F w C J t 0 7 H e 8 W n d b 7 b FIG. 1. Velocity-dependent DM elastic scattering cross sections (left) and GW spectra (right) for our benchmark points.See the text for details.

TABLE I .
Relevant parameters for the benchmark points of our model.
, and by the Key Research Program of the Chinese Academy of Sciences, grant No. XDPB15.CH acknowledges support from the Sun Yat-Sen University Science Foundation and the Fundamental Research Funds for the Central Universities, Sun Yat-sen University under Grant No. 23qnpy58.
Appendix A: Detailed expressions of thermal potential